Continuous
Limits and Intro to Derivatives
Bren Calculus Workshop
Carmen Galaz García, Ph.D.
Bren School of Environmental Science & Management
Last updated: Sep 10, 2025
Materials have been adapted and expanded from Nathaniel Grimes work for the Bren Calculus Workshop.
Team Review
How did everyone feel about the problem set?
Any questions?
Discuss with study partner
Functions are like baking recipes
In a function, these are the independent variables like \(x\).
These are the mathematical instructions in the function about how to manipulate the independent variables.
This is the ouput \(f(x)\) or dependent variable.
Typically we use the notation \(f(x)=x\), but we can always use different representations like \(g(x)=x\) or \(y=x\).
What is a function?
For each combination of independent variables, there is exactly one (and only one) value of the dependent variable.
Example
A function \(f(x)\) cannot return two values for \(x=1\): \(f(1)\ne\{2,3\}\)
Vertical line test
If you run a vertical line perpendicular to the \(x\)-axis and see where it intersects the graph of a function, then it should only intersect it once at every value of \(x\).
🤔 Are these functions?
Continuity
Intuitively, a function is continuous if there are no ‘breaks’ in the graph of the function. This means you are able to draw the whole graph without lifting your pencil.
To define it formally we need to understnad limits.
A function is discontinuous if it’s not continuous! So there are ‘breaks’ or ‘gaps’ in its graph.
Continuous
Discontinuous
Limits
Definition
We say a number \(L\) is the limit of a function \(f(x)\) as its variable \(x\) approaches a number \(c\), if the function’s output values \(f(x)\) approach \(L\) when \(x\) get closer to \(c\).
We write this symbolically as:
\[ \large \lim_{x\to c} f(x)=L \]
We read this as: “The limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\).”
Example: numerical
What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?
We need to examine the values of \(f(x)\) as \(x\) gets closer to 2 from both sides.
From both directions it looks like \(f(x)\) converges to 4.
So, we say that “The limit of \(2x^2-4\) as \(x\) approaches 2 is 4” and we can write it like this:
\[ \lim_{x\to 2}(2x^2-4)=4 \]
Example: graph
What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?
From both directions it looks like \(f(x)\) converges to 4.
So, \(\lim_{x\to 2}(2x^2-4)=4\).
Continuity revisited
Finding limits of functions when f(x) is continuous at c is easy.
\[ \lim_{x\to c}f(x)=f(c) \]
What happens with discontinuous functions?
Try with your group the same approach for:
\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]
Hint: Start very close to 2 like 1.9 and 2.1
We broke the universe
Dividing by zero is impossible
The function approaches different values from either side
Therefore…
Clearer on a graph
Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)
Defined Limit at an undefined point
Discontinuous functions can still have limits
Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2
\(x=2\) may not exist, but we can still find a limit because it consistently approaches 3 from both directions
Work with your team to complete these task
Part 1:
\[ \lim_{x\to3}g(x)=0 \]
\[ \lim_{x\to-4}g(x)=3 \]
\[ \lim_{x\to 3} g(x)= \text{DNE} \]
Work with your team to complete these task
Part 2:
Can you think of examples where discontinuous functions might exist in environmental science?
Choose as a team to draw an example of one of these statements:
\(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite
\(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)
\(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)
\(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)
Putting it all together
Recall average rate of change and instantaneous
Taking the average rate of change to a set limit will eventually converge to the instantaneous.
Walk Through Example
Walk Through Example
Walk Through Example
Walk Through Example
Walk Through Example
Walkthrough
Tangent Lines
What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).
These are called tangent lines
Put our example in math notation
\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]
Choose a point along the function \((x,f(x))\)
Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)
Add these into the slope equation
\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]
Derivative Definition
\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]
Most common notation
\[ \large f'(x)\text{, or }\frac{dy}{dx} \]
Not all functions are differentiable
All differentiable functions are continuous, but not all continuous functions are differentiable
The absolute value function is one example \(y=|x|\)
Calculus and Derivatives are the study of change
Environmental Science is also a study of change
Environmental Science is also a study of change
Environmental Science is also a study of change
Rules for Differentiation
Constant Rule
\[
\begin{align}
&y=a &\frac{dy}{dx}=0
\end{align}
\]
Power Rule
\[ \frac{d}{dx}[x^n] =nx^{n-1} \]
Examples
\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]
Rules for Differentiation
Sum and Difference Rules
\[
\begin{align}
\frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\
\frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x)
\end{align}
\]
Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.
Example
\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]
\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]
\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]